First, take your piece of cardboard and tear it into a size that matches up with your hole.

Cover the cardboard with mesh tape.

Use your putty knife to spread patching plaster or joint compound over the mesh tape. You want to be sure to completely cover the mesh tape, and you may need to do multiple layers to do so. If you are covering a hole in a door (as shown), you will want to scrape in the direction of the grain of the wood.

That brings you to this stage. Now, you can sand down the plaster to make it smooth. If you’ve been patching a wall, you will want to sand it down smooth and then imitate the texture used on the existing wall. (There are various textures in a can that you can buy and Lowe’s or Home Depot.)